Question

1. Weather Forecast

Jon and Ygritte plan to go hiking at a nearby trail in the North tomorrow morning. Unfortunately, the weatherman has predicted rainfall for tomorrow. When it actually rains, the weatherman correctly forecasts rain 90% of the time. When it doesn’t, he incorrectly forecasts rain 10% of the time. For all the following questions, first define the events for which you are calculating the probability, then calculate each of the probabilities you need to answer the question.

(a) Without any historic data, they assume that both rainfall and no rainfall have 50% chance on any given day. What is the probability that it will rain on the day of the hike, given that the weatherman predicted rain?

(b) Their initial estimation can be improved. They observe that in recent years, it has rained on average for 255.5 days each year. Using this new information, what is the probability that it will rain on the day of the hike, given that the weatherman predicted rain?

(c) Jon and Ygritte know that they can hike safely on the trail if the rainfall amount is not more than 20mm. So they collect daily rainfall data for the past year and learn that:

• On 30% of the days, it did not rain

• On 20% of the days, it rained 10mm

• On 20% of the days, it rained 20mm

• On 30% of the days, it rained 40mm

Using this information, answer the questions below:

i. If they do not listen to the weather forecast, what is the expected amount of rainfall on a given day?

ii. If they listen to the weather forecast, and rain is expected, what is the expected amount of rainfall on that day?

Answer 1

(a)

Let R and NR be the event of rain and no rain respectively.

Let F and NF be the event that the weatherman has correctly or not correctly forecasted the weather.

Given, P(F | R) = 0.90 P(F | NR) = 0.10

P(R) = P(NR) = 0.50

By law of total probability,

P(F) = P(R) P(F | R) + P(NR) P(F | NR)

= 0.5 * 0.90 + 0.50 * 0.10

= 0.5

Probability that it will rain on the day of the hike, given that the weatherman predicted rain = P(R | F)

= P(F | R) P(R) / P(F) (By Bayes theorem)

= 0.90 * 0.5 / 0.5

= 0.90

(b)

P(R) = 255.5 / 365 = 0.7

P(NR) = 1 - P(R) = 1 - 0.7 = 0.3

By law of total probability,

P(F) = P(R) P(F | R) + P(NR) P(F | NR)

= 0.7 * 0.90 + 0.3 * 0.10

= 0.66

Probability that it will rain on the day of the hike, given that the weatherman predicted rain = P(R | F)

= P(F | R) P(R) / P(F) (By Bayes theorem)

= 0.90 * 0.7 / 0.66

= 0.9545

(c)

(i)

Expected amount of rainfall on a given day = 0.30 * 0 + 0.2 * 10 + 0.2 * 20 + 0.3 * 40 = 18 mm

(ii)

Probability of rain = 0.7

Given,

P(Rained 10 mm | R) = P(Rained 10 mm and R) / P(R) = (Rained 10 mm) / P(R) = 0.2 / 0.7 = 2/7

Similarly,

P(Rained 20 mm | R) = P(Rained 20 mm and R) / P(R) = (Rained 20 mm) / P(R) = 0.2 / 0.7 = 2/7

P(Rained 40 mm | R) = P(Rained 40 mm and R) / P(R) = (Rained 40 mm) / P(R) = 0.3 / 0.7 = 3/7

Expected amount of rainfall on that day = (2/7) * 10 + (2/7) * 20 + (3/7) * 40 =25.71 mm